On the Torsion Congruence for Zeta Functions of Totally Real Fields

TL;DR

This paper investigates the congruences of special values of zeta functions for totally real fields using Shintani's cone decomposition, establishing results that support the noncommutative Iwasawa main conjecture.

Contribution

It proves a new congruence between zeta function values under prime degree extensions, supporting the torsion congruence in Iwasawa theory.

Findings

Established a congruence between zeta function special values for field extensions.

Connected the congruence to the torsion congruence in Iwasawa theory.

Supported the proof of the noncommutative Iwasawa main conjecture.

Abstract

In this note, we study the special values for zeta functions of totally real fields using the Shintani's cone decomposition. We prove certain congruence between the special values for zeta functions under the prime degree field extension. This congruence implies the `torsion congruence' proved by Ritter-Weiss which is crucial in the proof of the noncommutative Iwasawa main conjecture for totally real fields.